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Hole-assisted fibers have been proposed for a number of applications, including low-bend-loss access transmission. Suppression of higher-order modes is essential in these designs, and is explained here as the result of index-matched coupling between core and cladding modes. This physical principle is shown to explain previous empirically optimized designs, and enables intuitive generalizations. The improved tradeoff between bend loss and suppression of higher-order modes in these designs is discussed. Novel solid and microstructure fiber designs with suppressed higher-order modes illustrate these principles.
©2005 Optical Society of America
1. Introduction
Microstructure fibers have stimulated much recent research and have great potential in telecommunications and other applications [1, 2]. This potential is clearest when the unique properties of microstructure fiber can be directly related to qualitatively new functionality or dramatic differences in performance. For example, the ability to guide light in a low-index gas or liquid suggests a new regime of transmission with ultra-low nonlinearities [3, 4]. In addition, microstructure fibers may have more subtle but important advantages even when they behave more or less like standard fibers.
Hole-assisted designs have been proposed for low bend loss [5, 6], as well as other applications [7, 8]. Some of these designs have enhanced suppression of higher-order modes (HOM) compared to standard high-contrast fibers, but the physics behind this mode-suppression has not been fully explored.
In this paper, we show that suppression of HOM in hole-assisted fibers can be explained by a mechanism of index-matched coupling between core and cladding modes. This basic mechanism has been observed previously in high-dispersion fibers [9]. We then take this intuition and propose general design principles for HOM-suppressed fibers, both microstructure and solid. The basic design tradeoff between bend loss and HOM supression is explored for families of solid and hole-assisted fibers. We show that solid fibers can be designed to give quite similar behavior to hole-assisted designs, although practical differences remain.
2. Design tradeoffs: bend loss and single-modedness
There is a fundamental tradeoff in the design of low-bend loss fibers, depicted in Fig. 1. On the one hand, bending induces coupling to radiation, the evanescent leakage of power out of the core. Radiation loss can be reduced simply by increasing the index contrast of the core, thus confining light more strongly, or putting a larger barrier to leakage. However, when the contrast is too high, higher-order modes emerge, and can lead to increased bend sensitivity, multi-path interference, and other performance degradations. This basic tradeoff is somewhat similar to the one seen in large-mode-area fibers, where microstructure has generated interest [10, 11, 12]. One basic design strategy is simply to use a “just-right” index contrast, large enough to achieve a well-confined fundamental mode, and small enough to avoid penalties due to higher-order modes.
Figure 2 illustrates a more complex design strategy: Here, higher-order modes of a high-contrast core are suppressed by index-matched coupling between core and cladding modes. That is, a higher-index core provides a well-confined fundamental mode, and the accompanying unwanted higher-order modes are suppressed by including leaky guidance in the cladding. This paper shows that this is in fact the right description for understanding and designing the properties of HOM-suppressed fibers of the type proposed by Hasegawa et. al. [5]. Suppression of unwanted core modes occurs because these core modes resonantly mix with very leaky cladding modes when an index-matching condition is satisfied. This resonance condition explains intuitively why sharp optima were measured in the HOM loss as a function of geometrical parameters [5]. Similar coupling between a core mode and a ring mode in the cladding was explored previously for the purpose of generating large dispersion[13, 9]. Understanding this intuitive mechanism is very helpful in generalizing the design strategy, as we will see below.
Fig. 1. A simple step-index fiber design can achieve reduced macrobending losses by using higher index contrast (left). However, if contrast is too large (right), higher-order modes are guided in the core, leading to problems with microbend coupling.
Fig. 2. Cladding features can be designed to give index-matching between cladding modes and unwanted core modes. The fiber index profile is shown schematically with two high-index, waveguiding regions (core and cladding). The modes of the individual waveguides are represented by red curves (with effective index indicated by dashed lines). Index-matching leads to large mixing of energy between the unwanted modes and the cladding. At the same time, signal modes are not index-matched and have essentially no energy in the cladding region.
The design concept is not particular to hole-assisted fibers; in fact, it is straightforward to design solid, holey, or hole-assisted structures with the appropriate index matching between cladding modes and unwanted core modes. For a number of basic geometries, parameters can be tuned so that the effective index of cladding modes matches that of the unwanted modes at the operating wavelength, as shown in Fig. 3. The simulation results below were obtained using the multipole method for microstructure fibers [14] and the transfer-matrix method for radially symmetric fibers [15].
Fig. 3. Index matching of unwanted and cladding modes typically occurs over a narrow wavelength range. If the mode indices are plotted as a function of wavelength, the signal operating wavelength should be near the intersection of index curves for the unwanted and cladding modes.
3. Analysis of previous hole-assisted designs for low bend loss
Low bend loss microstructure fiber designs have been discussed previously [5, 6]. In all cases, very low bend loss levels are achieved by a very simple mechanism: large contrast between the up-doped core index and the effective cladding index (effectively down-doped by the air holes). Of course, achieving low bend loss with high-contrast fibers is not new in itself. The novelty of this work is that high contrast is achieved without compromising other design requirements. One design in particular [5] has two interesting features: 1. By adiabatically closing holes during a splice, an efficient mode-converter is built into the fiber design. The fiber can thus simultaneously be high-contrast for low bend loss (where the holes are open) and low-contrast for efficient splicing to SMF (at the splice, where the holes close). 2. By using a microstructure cladding with two different hole spacings, and tuning the spacing of the outer holes, HOM suppression is achieved. The first feature is an interesting potential advantage of microstructure fibers, and could be generalized to the mode-coupling needs of other applications. It is one of several proposed schemes for efficiently coupling to microstructure fibers [16, 17], which are currently being assessed based on cost and ease of implementation. In this paper, we focus not on this input coupling issue, but on the second feature: HOM suppression. Finally, the wavelength dependence of microstructure fibers can be very different from solid fibers, and may be an important factor distinguishing specific designs.
3.1. Optimizing the L1-L2 fiber
Consider the two-ring hole-assisted geometry (Fig. 4) explored previously [5], with hole spacings L 1 and L 2 for the first and second rings of holes, respectively. At first glance, it is not clear that there is any high index “ring” region in the cladding. However, an index-guiding ring emerges when we consider the effective index model of the fiber for the regular lattice case, L 2=L 1, and how this effective index might generalize when L 2 is made larger than L 1. The standard construction for an effective index, shown in Fig. 5, first replaces each unit cell in the lattice with an equivalent average index, and then approximates the resulting geometry with
a radially symmetric fiber with similar dimensions. While not rigorous, this has proven to be a fairly accurate and extremely useful construction [18, 19, 11]. Figure 6 shows one natural extension of this construction to the case where the outer hole spacing is increased; the space that opens between the two layers of holes can naturally be viewed as a high-index ring. We will show below that the full simulations bear out the “equivalent” core-ring fiber model as a useful approximation to the L 2>L 1 fiber.
Fig. 4. A two-ring hole-assisted fiber was proposed by Hasegawa, et. al., to achieve low bending loss. It consists of a central doped core surrounded by two hexagonal rings of air holes. Unequal hole spacings, L 1 and L 2, are used for the two rings in order to suppress the higher-order modes of the core.
Fig. 5. The effective index model is a useful approximation, where a periodic cladding is replaced by a single, wavelength-dependent average index. The appropriate index can be calculated using Bloch analysis (for example, see [18, 14]).
Further, we will see that it is precisely the leaky cladding modes guided by this ring that are responsible for the suppression of unwanted core modes. The mechanism of index-matched core-cladding coupling is illustrated well by the mode structure as the ratio L 2/L 1 is varied. Figure 7 (left) shows the periodic case, L 2=L 1. The fiber parameters follow Hasegawa, et. al. [5], with inner hole spacing L 1=7 microns, doped-core radius R core=4.3 and normalized dopant parameter Δcore=.0034. The natural categorization of “core” and “cladding” modes has been discussed previously [20, 21]. Core modes have n eff above the effective cladding index, and have power well confined to the core, while cladding modes have n eff below the cladding index, and power spread across the microstructured cladding [21].
Fig. 6. A high-index ring emerges when an effective-index approximation is used for the hole-assisted fiber with two hole spacings. The standard average-index concept can be generalized by averaging the index in a “unit-cell” about each hole, corresponding to the spacing in that cladding region.
Fig. 7. Numerical analysis of the hole-assisted fiber with regular-lattice cladding (left) shows core and cladding modes. Intensity plots are given for three modes indicated by black circles on the effective index vs. wavelength plot. Dashed lines and shading are used to highlight the doped-core index, the index of the undoped silica substrate, and the average cladding index obtained from Bloch analysis of the inner holes (dashed line at the bottom of the unshaded region). Core modes, both fundamental and higher-order, have power primarily in the core and effective index above the average-cladding index. The lower red lines represent cladding or “ring” modes, with power mostly outside the core. The ring mode index crosses above the average cladding index as the geometry is adjusted from L 2=L 1 (left) to L 2=1.05L 1 (right).
Fig. 8. When L 2/L 1 is chosen just right, core HOMs and cladding modes are index-matched and power mixes between the two regions (right, effective index and intensity plots). Loss is plotted (right) vs. wavelength for a few higher-order modes (the fundamental mode, in principle, has zero confinement loss). The loss peak near 1500nm wavelength can be interpreted as resonant HOM suppression due to coupling with the very leaky cladding modes.
As L 2/L 1 is increased, the core modes show little change in their effective index. The cladding modes (or “ring” modes), show a clear increase in effective index, above the average cladding index (dashed line, indicating the Bloch analysis for the first layer of holes). This is seen in Fig. 7 (right) for L 2/L 1=1.05. As L 2/L 1 is increased further, the index of the ring modes increases and can be made to match the index of the unwanted higher-order core modes. For this example, this point was reached for around L 2/L 1=1.2. The resulting mode structure shown in Fig. 8 (left) is now quite different: in place of the higher-order core modes and the ring modes, we see mixed core-ring modes, with a significant amount of power in each region. Because the ring modes for this structure suffer very high confinement losses, this mixing is sufficient to cause high loss of the unwanted higher-order core modes. Confinement losses are plotted in Fig. 8 (right), assuming a straight fiber. We then see that HOM suppression—discussed previously as a result of bending loss—is present even without bending the fiber. Confinement losses for the straight fiber can explain the observed higher-order mode suppression fairly accurately. Of course, models including bending may provide greater accuracy if the expected mechanical characteristics are known. Beyond the resonant-coupling point (L 2/L 1>1.2), the modes again resemble “pure” core and cladding modes at wavelengths around 1500nm, as shown in Fig. 9.
A bend-resistant fiber design requires understanding the tradeoff between macro-bending loss (of the fundamental) and HOM suppression. Interestingly, one can get a qualitative feel for this tradeoff simply from the straight-fiber mode analysis. Fig. 10 summarizes the results of such an analysis, where macrobending sensitivity was estimated using a crude bending loss proxy. For the example in question, this approach is quite successful (compare with [5, Fig 2]). While a more rigorous bending-loss analysis may still be desirable, simplified estimates would potentially be very useful in rapid analysis and design. The bending loss proxy is simply the radius at which the intensity falls to a fixed, small fraction of its peak value. The construction is shown graphically in Fig. 11, where the intensity is averaged over azimuthal angle, and the radius R is defined by I avg(R)/I(0)=10-7. This bending loss proxy is reasonable as a general indicator of how well confined the fundamental is, and has some correspondence to more correct expressions for bending loss. Further experimental data will be needed to see whether the proxy is a useful indicator for more general designs.
Fig. 9. When L 2/L 1>1.2, the core and cladding modes are again not index-matched (in the 1500nm region), and relatively little mixing is seen.
Fig. 10. The basic tradeoff between macrobending loss and higher-order mode suppression can be summarized using a bend-loss proxy vs. HOM loss plot if there is a single wavelength of interest. Here the stars indicate the L 2=L 1 designs, with colors indicating different hole diameters. Fitting the stars, we can infer the basic tradeoff possible using only hole size as a degree of freedom (black dashed line). Colored lines show the improved performance as the additional degree of freedom L 2>L 1 is tuned.
Interestingly, Fig. 10 confirms the basic design strategy underlying the L 1-L 2 fiber family: that one can simultaneously improve both the macrobending loss and the HOM suppression relative to a periodic (L 2=L 1) design. The periodic designs (L 2=L 1) are indicated by stars fit by a dashed black line, and the solid curves (L 2>L 1) generally fall above and to the left of the stars. This shows that tuning L 2 improves both (generally conflicting) requirements: fundamental confinement and HOM suppression.
Fig. 11. The plot of intensity as a function of radius indicates the degree of confinement of the fundamental mode, and should correlate with resistance to macrobending losses. A bend-loss proxy is defined as the radius at which the intensity reaches a fixed threshold (for example 10-7) times its peak value (red dashed line). Light blue curves include intensity at several angles, while the dark blue curve indicates the average over azimuthal angle. Red star, circle, and triangle symbols mark the radius of intersection for three fiber geometries. As expected, the hole-assisted fibers have a more tightly confined mode than the step-index fiber with identical core but no holes (dashed black line).
If a single wavelength of operation is defined, then the basic design tradeoff is summarized by a plot such as Fig. 10. However, if a broad range of wavelengths are of interest, it is better to consider the HOM suppression over this entire range. We see in Figs. 8 (right) and 12 (left) that resonant coupling can lead to a very wavelength-sensitive loss. One useful parameter might be the “effective cutoff” wavelength, defined as the wavelength above which HOMs are adequately suppressed. This “cutoff” can vary substantially depending on the fairly arbitrary threshold defining the adequate HOM confinement, as indicated by red stars in Fig. 12 (left). The right plot of Fig. 12 gives an example of a broadband design-tradeoff summary for the L 1-L 2 fiber. Like Fig. 10, this summary confirms that the “tuned” L 2>L 1 designs can simultaneously improve both the macrobending loss and the HOM suppression relative to the more restricted, periodic (L 2=L 1) class of designs. Solid curves (L 2>L 1) fall below and to the left of the periodic designs (indicated by stars), indicating better (shorter) cutoff for a given bend loss proxy.
Resonant coupling requires fairly accurate tuning of the structure. It is certainly not an automatic property of hole-assisted fibers; in fact, we should expect that most of the possible hole-assisted design space follows essentially the same tradeoffs between mode-confinement and higher-order modes as solid doped fibers, and will not have any special mode-suppression properties. Conversely, we should not assume that HOM-suppression designs can only be achieved using hole-assisted fibers. In the next section, we discuss solid fiber designs with essentially the same mode-suppression behavior as the L 1-L 2 designs.
Fig. 12. For broadband applications, an effective cutoff wavelength summarizes the HOM confinement over a wavelength range into a single number. A cutoff can be extracted simply by thresholding (left), and will depend on the value of the loss threshold (probably empirical). The basic design tradeoff can then be summarized in a plot of macrobending loss vs. cutoff (right). All cutoffs that are out-of-range are plotted as 1800nm.
4. Generalizing HOM-suppressed designs
We saw in Fig. 6 that the notion of an effective cladding index could be generalized to give rough solid-fiber “equivalents” to a microstructure fiber. This can serve as a simplified model for analysis, or as a good starting point in finding useful solid-fiber designs. The strategy of mode suppression through core-cladding coupling is very general, and can be achieved with many different designs, but it is instructive to start with these solid-fiber “equivalents,” to highlight similarities and differences.
4.1. Doped-fiber equivalents
A family of radial fiber designs were constructed with the “L 1-L 2-equavalent” index profile:
Bloch analysis gives n holes1 as the average index of a silica-air lattice with hole spacing L 1, diameter d, and wavelength λ=1550nm. Similarly n holes2 corresponds to hole spacing L 2 of the outer ring. Given the qualitative nature of the “equivalent” fiber, the precise details of the construction should not be essential to the behavior. For simplicity, all of the material indices were taken to be constant, with silica index n sil=1.444. For the particular L 1-L 2 fiber used (R core=4.3 and n core=1.0034n sil.), R core>L 1/2 and so we used a modified core forced to fit inside of a “unit cell” of size L 1. Specifically, we rescaled the core index contrast
As expected, this construction gave a solid fiber whose basic mode structure, bend-loss proxy, and HOM confinement loss are qualitatively quite similar to the associated family of L 1-L 2 fiber designs. The agreement is not quantitative (not surprising considering the crude nature of the construction, Fig. 6). Figure 13 confirms that mixing between core and ring modes near the index-matching point is again very relevant at wavelengths of interest.
Fig. 13. Solid fiber designs were constructed as crude equivalents to the L 1-L 2 family of hole-assisted fibers. Mode analysis shows that mixing of core and ring modes arises and leads to suppression of unwanted higher-order modes. Intensity plots show poorly confined higher-order modes near the index-matching wavelength. Dashed black lines and shading indicate the four material index values of the fiber (top to bottom): ñcore, n sil=1.444, n holes2, and n holes1.
Figure 14 shows two summary plots for the family of designs, indicating the tradeoff between fundamental mode confinement and HOM suppression. As before, we quantify HOM suppression in two different ways: a confinement loss value indicates narrowband performance at 1550nm (left), and a cutoff estimate is used to summarize confinement losses over a broad wavelength range of interest (right). As before, the summary plots confirm the basic design strategy: Tuning the index-matched HOM suppression is a better way to simultaneously improve both metrics than changing index contrast alone. Compared to the simpler core-trench designs (L 2=L 1, without a raised-index ring, indicated by stars) the tuned designs (L 2>L 1, shown as solid lines) can simultaneously keep the fundamental better confined and make the HOMs more leaky.
Another interesting result of this analysis is the influence of trench depth (parameterized by hole diameter d, or lower trench index for solid fibers). Figures 10 and 14 confirm that there is an optimum L 2/L 1 ratio for a fixed wavelength and hole diameter, but also show that this optimum becomes sharper as the trench becomes deeper. Physically, this is because the cladding mode index becomes more sharply defined as the cladding modes become less leaky. Practically, it means that making the holes too large can be problematic. Designs with large d are more sensitive to precise tuning of L 2/L 1 and have overall lower HOM suppression. On the other hand, very small trench depths (or holes) will simply reduce to a step-index fiber. One would expect moderate trench depths to perform better than depths that are too big or too small.
We note that the equivalent HOM suppression of the solid fiber designs does not imply that they are equivalent in all respects to hole-assisted counterparts. Above we have mentioned the potential for mode conversion in a tapered hole-assisted fiber. While practical issues (coupling, birefringence, cost, etc.) will require ongoing investigation, it is interesting to see that the unusual HOM-suppression feature is not unique to fibers with holes.
Fig. 14. The basic design tradeoff between a well-confined fundamental (low macrobending loss) and poorly-confined higher-order modes is summarized by simple numerical estimates for the solid L 1-L 2-equivalent fiber family. The presence of cladding rings (solid curves) simultaneously improves both performance metrics. Performance at a specific wavelength is summarized as HOM loss vs. bend-loss proxy (left), where desirable designs have a low proxy and high HOM loss. Broadband performance is instead summarized as bend loss proxy versus cutoff (right), with low cutoffs being desirable. Stars represent the special case L 2=L 1, where there is no high-index ring.
Fig. 15. This hole-assisted fiber has holes fairly evenly distributed through the cladding. The effective index of the cladding is roughly .01 below that of the substrate, so that the first higher-order mode group is well within the unshaded hole-confined region, indicating low-loss confinement in the core.
4.2. Other hole-assisted designs
As another example, consider a hole-assisted fiber with superficially very different cladding parameters: Figure 15 shows a doped high-index core surrounded by a cladding with circular but roughly uniform placement of holes. The three layers of holes have 12, 18, and 24 holes and radius 7, 10.5, and 14 microns. All hole diameters were 1 micron. Mode effective index are plotted as colored lines, while the lower dashed black line shows the average cladding index. The hole size and spacing were chosen to give an effective index contrast on the same order as the doped-core contrast. The core supports a fundamental mode (upper green curve), one well-confined group of HOM modes (upper red curve), and some high-loss modes near the edge of the confinement region. We would like to suppress this highest-index (low loss) HOM group, since it presents the dominant multi-path interference impairment.
Fig. 16. A hole-assisted fiber with a missing layer of holes has a high-index ring in the effective index profile. Index matching between the cladding modes of the ring and higher-order modes of the core is achieved at around 1.55 microns, as seen in the effective index plot (left). The loss (right) near this wavelength is hundreds of times larger than the loss of the corresponding three-layer HAF with no index-matched cladding modes (shown dashed). For this fiber, the core diameter is 7.45 microns with 0.38% relative index difference. The cladding has 1 micron diameter holes, with 12-holes on a circular layer of radius 7 microns and 24 holes on a second circular layer of radius 14.7 microns. For reference, the dashed curve gives the loss of the Fig. 15, where no holes are “missing.” Blue curves are used for modes with the same “even” symmetry as the fundamental, and red curves have “odd” symmetry. The index-matched loss curves are highlighted with black.
An HOM-suppressed version of this design requires a high effective-index ring where holes are missing. Specifically, we remove the second layer of cladding holes, and then fine tune the structure by changing the radius of the third layer. The effective index versus wavelength plot, Fig. 16 (left), shows little change in the fundamental mode, but many cladding modes have been introduced in the unshaded region. Simulations show a very well-confined fundamental and a great enhancement of the higher-order mode loss, as desired [Fig. 16 (right)]. The peak of the HOM loss (related to the index-matching point) was successfully tuned by when the radius of the outer layer of holes was adjusted (the layer radius is 14.7 microns for the figure; other adjustments are not shown).
It is worth noting that, while the minimum HOM loss is a fairly good indicator of HOM suppression, the relevant impairments may not exclusively depend on a single higher-order mode. For example, Figs. 16 (right) and 8 (right) show that the 36-hole fiber has many higher-order modes with fairly similar loss values, of order dB/mm, unlike the L 1-L 2 fibers. If several modes (including the index-matched ones we have discussed) have similar losses, they will participate in microbend coupling and may have a significant impact on fiber impairments. The number of HOMs as well as their losses may favor some fiber designs. Future modeling and measurements are needed to determine exactly how transmission impairments relate to fiber design, and what limits on impairments are required by applications.
5. Conclusions
A general design principle of mode suppression through index-matched mixing with leaky cladding modes has been identified in this paper. We have shown that this resonant suppression mechanism can selectively suppress unwanted higher-order modes while maintaining good confinement of desirable fundamental modes. Our analysis suggests that this is the dominant mechanism underlying the HOM suppression in a family of existing low-bend loss fibers.
Simulations show that this design strategy can be used to quickly generalize from pre-existing designs to quite different solid or hole-assisted designs with similar underlying physics. The strategy can now be used to move towards designs that better address the needs of cutting-edge applications, such as fiber to the home.
Acknowledgments
I thank Siddharth Ramachandran, Ryan Bise, Dennis Trevor and David DiGiovanni for useful discussions. This work was supported in part by the NIST ATP program under award number 70NANB4H3035.
References and links
1. T. A. Birks and P. J. Roberts, et al. “Full 2d photonic bandgaps in silica/air structures,” Electron. Lett. 31, 1941 (1995). [CrossRef]
2. S. A. Diddams and D. J. Jones, et al. “Direct link between microwave and optical frequencies with a 300THz femtosecond laser comb,” Phys. Rev. Lett. 84, 5102 (2000). [CrossRef] [PubMed]
3. B. J. Mangan, L. Farr, A. Langford, P. J. Roberts, D. P. Williams, F. Couny, M Lawman, M. Mason, S Coupland, R. Flea, H. Sabert, T. A. Birks, J. C. Knight, and P. St. J. Russel. “Low loss (1.7db/km) hollow core photonic bandgap fiber,” In Optical Fiber Communications Conference (OFC), PDP24 (2004).
4. S. G. Johnson and M. Ibanescu, et al. “Low-loss asymptotically single-mode propagation in large-core OmniGuide fibers,” Opt. Express 9, 748 (2001). [CrossRef] [PubMed]
5. T. Hasegawa, T. Saitoh, D. Nishioka, E. Sasaoka, and T. Hosoya. Bend-insensitive single-mode holey fibre with SMF compatibility for optical wiring applications. In European Conference on Optical Communications, page We2.7.3 (2003).
6. K. Nakajima and K. Hogari, et al. “Hole assisted fiber design for small bending and splice losses,” Photon. Technol. Lett. 15,1737 (2003). [CrossRef]
7. T. Hasegawa, E. Sasaoka, M. Onishi, M. Nishimura, Y. Tsuji, and M. Koshiba. “Hole-assisted lightguide fiber for large anomalous dispersion and low optical loss,” Opt. Express 9, 681 (2001). [CrossRef] [PubMed]
8. M. Yan, P. Shum, and C. Lu. “Hole-assisted multiring fiber with low dispersion around 1550 nm,” Photon. Technol. Lett. 16, 123 (2004). [CrossRef]
9. F. Gerome, J. L. Auguste, and J. M. Blondy. “Design of dispersion-compensating fibers based on a dual-concentric-core photonic crystal fiber,” Opt. Lett. 29, 2725 (2004). [CrossRef] [PubMed]
10. N. A. Mortensen, M. D. Nielsen, J. R. Folkenberg, K. P. Hansen, and H. R. Simonsen. “Improved large-mode-area endlessly single-mode photonic crystal fibers,” Opt. Lett. 28, 393 (2003). [CrossRef] [PubMed]
11. M. D. Nielsen, N. A. Mortensen, M. Albertsen, J. R. Folkenberg, A. Bjarklev, and D. Bonacinni. “Predicting macrobending loss for large-mode area photonic crystal fibers,” Opt. Express 11, 2762 (2004). [CrossRef]
12. J. Limpert, A. Liem, M. Reich, T. Schreiber, S. Nolte, H. Zellmer, A. Tnnermann, J. Broeng, A. Petersson, and C. Jakobsen. “Low-nonlinearity single-transverse-mode ytterbium-doped photonic crystal fiber amplifier,” Opt. Express 12, 1313–1319 (2004). [CrossRef] [PubMed]
13. F. Gerome, J. L. Auguste, and J. M. Blondy. “Very high negative chromatic dispersion in a dual concentric core photonic crystal fiber,” In Optical Fiber Communications Conference (OFC), WA2 (2004).
14. T. P. White, R. C. McPhedran, L. C. Botten, G. H. Smith, and C. M. deSterke. “Calculations of air-guiding modes in photonic crystal fibers using the multipole method,” Opt. Express 9, 721 (2001). [CrossRef] [PubMed]
15. P. Yeh, A. Yariv, and E. Marom. “Theory of Bragg fiber,” J. Opt. Soc. Am. 68, 1196 (1978). [CrossRef]
16. A. D. Yablon and Ryan Bise. “Low-loss high-strength microstructure fiber fusion splices using grin fiber lenses,” In Optical Fiber Communications Conference (OFC), MF14 (2004).
17. J. K. Chandalia, B. J. Eggleton, R. S. Windeler, S. G. Kosinski, X. Liu, and C. Xu. “Adiabatic coupling in tapered air-silica microstructured optical fiber,” Photon. Technol. Lett. 13, 52 (2001). [CrossRef]
18. J. C. Knight, T. A. Birks, P. S. J. Russell, and J. P. de Sandro. “Properties of photonic crystal fiber and the effective index model,” J. Opt. Soc. Am. A 15, 748 (1998). [CrossRef]
19. T. M. Monro and D. J. Richardson, et al. “Holey optical fibers: an efficient modal model,” J. Lightwave Technol. 17, 1093 (1999). [CrossRef]
20. B. T. Kuhlmey, R. C. McPhedran, and C. M. deSterke. “Modal cutoff in microstructured optical fibers,” Opt. Lett. 27, 1684 (2002). [CrossRef]
21. J. M. Fini. “Bloch theory describing cladding modes of microstructure optical fiber,” In European Conference on Optical Communications, We4.P.37, (2003).
